1 / | | _________ | / 2 2 | \/ x - a dx | / 0
Integral(sqrt(x^2 - a^2), (x, 0, 1))
// 2 /x\ \ || a *acosh|-| 3 | 2| | || \a/ x a*x |x | | ||- ----------- + ------------------- - ----------------- for |--| > 1| || 2 _________ _________ | 2| | / || / 2 / 2 |a | | | || / x / x | | _________ || 2*a* / -1 + -- 2* / -1 + -- | | / 2 2 || / 2 / 2 | | \/ x - a dx = C + |< \/ a \/ a | | || | / || ________ | || / 2 | || / x | || 2 /x\ I*a*x* / 1 - -- | || I*a *asin|-| / 2 | || \a/ \/ a | || ------------ + -------------------- otherwise | \\ 2 2 /
1 / | | / 2 4 2 2 | | a x x 3*x x | |- --------------- + ---------------- - ----------------- + ------------------- for ---- > 1 | | _________ 3/2 3/2 _________ | 2| | | / 2 / 2\ / 2\ / 2 |a | | | / x | x | 3 | x | / x | | / -1 + -- 2*a*|-1 + --| 2*a *|-1 + --| 2*a* / -1 + -- | | / 2 | 2| | 2| / 2 | | \/ a \ a / \ a / \/ a | | | | ________ | < / 2 dx | | / x | | I*a* / 1 - -- | | / 2 2 | | \/ a I*a I*x | | ------------------ + ---------------- - ------------------ otherwise | | 2 ________ ________ | | / 2 / 2 | | / x / x | | 2* / 1 - -- 2*a* / 1 - -- | | / 2 / 2 | \ \/ a \/ a | / 0
=
1 / | | / 2 4 2 2 | | a x x 3*x x | |- --------------- + ---------------- - ----------------- + ------------------- for ---- > 1 | | _________ 3/2 3/2 _________ | 2| | | / 2 / 2\ / 2\ / 2 |a | | | / x | x | 3 | x | / x | | / -1 + -- 2*a*|-1 + --| 2*a *|-1 + --| 2*a* / -1 + -- | | / 2 | 2| | 2| / 2 | | \/ a \ a / \ a / \/ a | | | | ________ | < / 2 dx | | / x | | I*a* / 1 - -- | | / 2 2 | | \/ a I*a I*x | | ------------------ + ---------------- - ------------------ otherwise | | 2 ________ ________ | | / 2 / 2 | | / x / x | | 2* / 1 - -- 2*a* / 1 - -- | | / 2 / 2 | \ \/ a \/ a | / 0
Integral(Piecewise((-a/sqrt(-1 + x^2/a^2) + x^2/(2*a*(-1 + x^2/a^2)^(3/2)) - x^4/(2*a^3*(-1 + x^2/a^2)^(3/2)) + 3*x^2/(2*a*sqrt(-1 + x^2/a^2)), x^2/|a^2| > 1), (i*a*sqrt(1 - x^2/a^2)/2 + i*a/(2*sqrt(1 - x^2/a^2)) - i*x^2/(2*a*sqrt(1 - x^2/a^2)), True)), (x, 0, 1))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.